2003 AMC 12B Problem 20

Problem 20 of 25HarderAlgebra

Part of the graph of

f(x) = ax^3 + bx^2 + cx + d

is shown. What is

b?

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Solution

The graph passes through

(-1, 0),

(1, 0),

and

(0, 2).

f(0) = d = 2.

Adding

\begin{aligned} &f(1) + f(-1) \\ &= (a + b + c + d) \\ &\quad {}+ (-a + b - c + d) \\ &= 2b + 2d = 0, \end{aligned}

b = -d = -2.

Thus, the correct answer is B.

Concepts: polynomial · substitution

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.